Advertisement

Journal of Global Optimization

, Volume 18, Issue 4, pp 301–320 | Cite as

On Copositive Programming and Standard Quadratic Optimization Problems

  • Immanuel M. Bomze
  • Mirjam Dür
  • Etienne de Klerk
  • Cornelis Roos
  • Arie J. Quist
  • Tamás Terlaky
Article

Abstract

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

Copositive programming Global maximization Positive semidefinite matrices Standard quadratic optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baum, L.E. and Eagon, J.A. (1967), An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology. Bull. Amer. Math. Soc. 73: 360-363.Google Scholar
  2. 2.
    Baum, L.E. and Sell, G.R. (1968), Growth transformations for functions on manifolds. Pacif. J. Math. 27: 211-227.Google Scholar
  3. 3.
    Berman, A. (1988), Complete positivity. Linear Algebra Appl. 107: 57-63.Google Scholar
  4. 4.
    Berman, A. and Hershkowitz, D. (1987), Combinatorial results on completely positive matrices. Linear Algebra Appl. 95: 111-125.Google Scholar
  5. 5.
    Bomze, I.M. (1987), Remarks on the recursive structure of copositivity. J.Inf.Optimiz.Sci. 8: 243-260.Google Scholar
  6. 6.
    Bomze, I.M. (1996), Block pivoting and shortcut strategies for detecting copositivity. Linear Algebra Appl. 248: 161-184.Google Scholar
  7. 7.
    Bomze, I.M. (1997), Evolution towards the maximum clique. J. Global Optimiz. 10: 143-164.Google Scholar
  8. 8.
    Bomze, I.M. (1997), Global escape strategies for maximizing quadratic forms over a simplex. J. Global Optimiz. 11: 325-338.Google Scholar
  9. 9.
    Bomze, I.M. (1998), On standard quadratic optimization problems. J. Global Optimiz. 13: 369-387.Google Scholar
  10. 10.
    Bomze, I.M. (2000), Linear-time detection of copositivity for tridiagonal matrices and extension to block-tridiagonality. SIAM J. Matrix Anal. Appl.. 21: 840-848.Google Scholar
  11. 11.
    Bomze, I.M., Pelillo, M. and Giacomini, R. (1997), Evolutionary approach to the maximum clique problem: empirical evidence on a larger scale. In: Bomze, I.M., Csendes, T., Horst, R. and Pardalos, P.M. (eds.), Developments in Global Optimization, Kluwer, Dordrecht, pp. 95-108.Google Scholar
  12. 12.
    Bomze, I.M. and Stix, V. (1999), Genetical engineering via negative fitness: evolutionary dynamics for global optimization. Annals of O.R. 89: 279-318.Google Scholar
  13. 13.
    Crow, J.F., Kimura, M. (1970), An Introduction to Population Genetics Theory, Harper & Row, New York.Google Scholar
  14. 14.
    Danninger, G. (1990), A recursive algorithm to detect (strict) copositivity of a symmetric matrix. In: Rieder, U., Gessner, P., Peyerimhoff, A. and Radermacher, F.J., (eds.), Methods of Operations Research 62: 45-52. Hain, Meisenheim.Google Scholar
  15. 15.
    Drew, J.H., Johnson, C.R. and Loewy, R. (1994), Completely positive matrices associated with M-matrices. Linear and Multilinear Algebra 37: 303-310.Google Scholar
  16. 16.
    Fujisawa, K., Kojima, M. and Nakata, K. (1997), Exploiting sparsity in primal-dual interiorpoint methods for semidefinite programming. Math. Prog. 79: 235-253.Google Scholar
  17. 17.
    Fukuda, M., Kojima, M., Murota, K. and Nakata, K. (1999), Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework, Research Report B-358, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan.Google Scholar
  18. 18.
    Goemans, M.X. (1997), Semidefinite programming in combinatorial optimization. Math. Prog. 79: 143-161.Google Scholar
  19. 19.
    Hadeler, K.P. (1983), On copositive matrices. Linear Algebra Appl. 49: 79-89.Google Scholar
  20. 20.
    Hall, M., Jr. and M. Newman, (1963), Copositive and completely positive quadratic forms. Proc. Cambridge Philos. Soc. 59: 329-339.Google Scholar
  21. 21.
    Hannah, J. and Laffey, T.J. (1983), Nonnegative factorization of completely positive matrices. Linear Algebra Appl. 55: 1-9.Google Scholar
  22. 22.
    Helmberg, C., Rendl, F., Vanderbei, R.J. and Wolkowicz, H. (1996), An interior-point method for semidefinite programming. SIAM J. Optimiz. 6: 342-361.Google Scholar
  23. 23.
    Hofbauer, J. and Sigmund, K. (1988), The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, UK.Google Scholar
  24. 24.
    Horst, R., Pardalos, P.M. and Thoai, N.V. (1995), Introduction to Global Optimization, Kluwer, Dordrecht.Google Scholar
  25. 25.
    Hummel, R.A. and Zucker, S.W. (1983), On the foundations of relaxation labeling processes. IEEE Trans. Pattern Anal. Machine Intell. 5: 267-287.Google Scholar
  26. 26.
    Jansen, B., Roos, C. and Terlaky, T. (1995), Interior point methods: a decade after Karmarkar. A survey, with application to the smallest eigenvalue problem. Statistica Neerlandica 50: 146-170.Google Scholar
  27. 27.
    Kaykobad, M. (1987), On nonnegative factorization matrices. Linear Algebra Appl. 96: 27-33.Google Scholar
  28. 28.
    Klerk, E. de, Roos, C. and Terlaky, T. (1998), Polynomial primal-dual affine scaling algorithms in semidefinite programming. J. Combin. Optimiz. 2: 51-69.Google Scholar
  29. 29.
    Kojima, M., Shindoh, S. and Hara, S. (1997), Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optimiz. 7: 86-125.Google Scholar
  30. 30.
    Kojima, M. and Tunçel, L. (1998), Monotonicity of primal-dual interior-point algorithms for semidefinite programming problems. Optimiz. Methods Softw. 10: 275-296.Google Scholar
  31. 31.
    Lau, C.M. and Markham, T.L., (1978), Square triangular factorizations of completely positive matrices. J. Ind. Math. Soc. 28: 15-24.Google Scholar
  32. 32.
    Levinson, S.E., Rabiner, L.R. and Sondhi, M.M. (1983), An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition. Bell Syst. Tech. J. 62: 1035-1074.Google Scholar
  33. 33.
    Lyubich, Y., Maistrowskii, G. D. and Ol'khovskii, Yu.G. (1980), Selection-induced convergence to equilibrium in a single-locus autosomal population. Problems of Information Transmission 16: 66-75.Google Scholar
  34. 34.
    Markham, T. L. (1971), Factorizations of completely positive matrices. Proc. Cambridge Philos. Soc. 69: 53-58.Google Scholar
  35. 35.
    Martin, D.H. and Jacobson, D.H. (1981), Copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints, Linear Algebra Appl. 35: 227-258.Google Scholar
  36. 36.
    Monteiro, R.D.C. (1997), Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optimiz. 7: 663-678.Google Scholar
  37. 37.
    Muramatsu, M. and Vanderbei, R.J. (1997), Primal-dual affine-scaling algorithm fails for semidefinite programming. Technical Report SOR 97-04, Princeton University, NJ 08544.Google Scholar
  38. 38.
    Murty, K.G. and Kabadi, S.N. (1987), Some NP-complete problems in quadratic and linear programming. Math. Prog. 39: 117-129.Google Scholar
  39. 39.
    Nesterov, Y.E. and Nemirovskii, A.S. (1994), Interior point methods in convex programming: theory and applications. SIAM, Philadelphia, PA.Google Scholar
  40. 40.
    Nowak, I. (1999), A new semidefinite programming bound for indefinite quadratic forms over a simplex. J. Global Optimiz. 14: 357-364.Google Scholar
  41. 41.
    Pelillo, M. (1994), On the dynamics of relaxation labeling processes. Proc. IEEE Int. Conf. Neural Networks, Orlando, FL, pp. 1006-1011.Google Scholar
  42. 42.
    Pelillo, M. (1995), Relaxation labeling networks for the maximum clique problem. J. Artif. Neural Networks 2: 313-327.Google Scholar
  43. 43.
    Raber, U. (1999), Nonconvex all-quadratic global optimization problems: solution methods, application and related topics. Ph.D. dissertation, University of Trier.Google Scholar
  44. 44.
    Rosenfeld, A., Hummel, R.A. and Zucker, S.W. (1976), Scene labeling by relaxation operations. IEEE Trans. Syst. Man Cybern. 6: 420-433.Google Scholar
  45. 45.
    Quist, A.J., de Klerk, E., Roos, C. and Terlaky, T. (1998), Copositive relaxation for general quadratic programming. Optimiz. Methods Softw. 9: 185-209.Google Scholar
  46. 46.
    Sigmund, K. (1987), Game dynamics, mixed strategies, and gradient systems. Theor. Pop. Biol. 32: 114-126.Google Scholar
  47. 47.
    Tunçel, L. (1999), Generalization of primal-dual interior-point methods to convex optimization problems in conic form. Technical Report CORR 99-35, University of Waterloo.Google Scholar
  48. 48.
    Väliaho, H. (1986), Criteria for copositive matrices. Linear Algebra Appl. 81: 19-34.Google Scholar
  49. 49.
    Väliaho, H. (1988), Testing the definiteness of matrices on polyhedral cones. Linear Algebra Appl. 101: 135-165.Google Scholar
  50. 50.
    Väliaho, H. (1989), Quadratic-programming criteria for copositive matrices. Linear Algebra Appl. 119: 163-182.Google Scholar
  51. 51.
    Xiang, S.H. and Xiang, S.W. (1998), Notes on completely positive matrices. Linear Algebra Appl. 271: 273-282.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  • Mirjam Dür
    • 2
  • Etienne de Klerk
    • 3
  • Cornelis Roos
    • 3
  • Arie J. Quist
    • 3
  • Tamás Terlaky
    • 4
  1. 1.ISDSUniversität WienAustria
  2. 2.Department of StatisticsVienna University of EconomicsAustria
  3. 3.Faculty ITS/TWI/SSORDelft University of TechnologyThe Netherlands
  4. 4.Department of Computing and SoftwareMcMaster University HamiltonCanada

Personalised recommendations